68,390 research outputs found

    Some Applications of Coding Theory in Computational Complexity

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    Error-correcting codes and related combinatorial constructs play an important role in several recent (and old) results in computational complexity theory. In this paper we survey results on locally-testable and locally-decodable error-correcting codes, and their applications to complexity theory and to cryptography. Locally decodable codes are error-correcting codes with sub-linear time error-correcting algorithms. They are related to private information retrieval (a type of cryptographic protocol), and they are used in average-case complexity and to construct ``hard-core predicates'' for one-way permutations. Locally testable codes are error-correcting codes with sub-linear time error-detection algorithms, and they are the combinatorial core of probabilistically checkable proofs

    Construction algorithm for network error-correcting codes attaining the Singleton bound

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    We give a centralized deterministic algorithm for constructing linear network error-correcting codes that attain the Singleton bound of network error-correcting codes. The proposed algorithm is based on the algorithm by Jaggi et al. We give estimates on the time complexity and the required symbol size of the proposed algorithm. We also estimate the probability of a random choice of local encoding vectors by all intermediate nodes giving a network error-correcting codes attaining the Singleton bound. We also clarify the relationship between the robust network coding and the network error-correcting codes with known locations of errors.Comment: To appear in IEICE Trans. Fundamentals (http://ietfec.oxfordjournals.org/), vol. E90-A, no. 9, Sept. 2007. LaTeX2e, 7 pages, using ieice.cls and pstricks.sty. Version 4 adds randomized construction of network error-correcting codes, comparisons of the proposed methods to the existing methods, additional explanations in the proo

    Stabilizer codes from modified symplectic form

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    Stabilizer codes form an important class of quantum error correcting codes which have an elegant theory, efficient error detection, and many known examples. Constructing stabilizer codes of length nn is equivalent to constructing subspaces of Fpn×Fpn\mathbb{F}_p^n \times \mathbb{F}_p^n which are "isotropic" under the symplectic bilinear form defined by (a,b),(c,d)=aTdbTc\left\langle (\mathbf{a},\mathbf{b}),(\mathbf{c},\mathbf{d}) \right\rangle = \mathbf{a}^{\mathrm{T}} \mathbf{d} - \mathbf{b}^{\mathrm{T}} \mathbf{c}. As a result, many, but not all, ideas from the theory of classical error correction can be translated to quantum error correction. One of the main theoretical contribution of this article is to study stabilizer codes starting with a different symplectic form. In this paper, we concentrate on cyclic codes. Modifying the symplectic form allows us to generalize the previous known construction for linear cyclic stabilizer codes, and in the process, circumvent some of the Galois theoretic no-go results proved there. More importantly, this tweak in the symplectic form allows us to make use of well known error correcting algorithms for cyclic codes to give efficient quantum error correcting algorithms. Cyclicity of error correcting codes is a "basis dependent" property. Our codes are no more "cyclic" when they are derived using the standard symplectic forms (if we ignore the error correcting properties like distance, all such symplectic forms can be converted to each other via a basis transformation). Hence this change of perspective is crucial from the point of view of designing efficient decoding algorithm for these family of codes. In this context, recall that for general codes, efficient decoding algorithms do not exist if some widely believed complexity theoretic assumptions are true

    Local Testing for Membership in Lattices

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    Motivated by the structural analogies between point lattices and linear error-correcting codes, and by the mature theory on locally testable codes, we initiate a systematic study of local testing for membership in lattices. Testing membership in lattices is also motivated in practice, by applications to integer programming, error detection in lattice-based communication, and cryptography. Apart from establishing the conceptual foundations of lattice testing, our results include the following: 1. We demonstrate upper and lower bounds on the query complexity of local testing for the well-known family of code formula lattices. Furthermore, we instantiate our results with code formula lattices constructed from Reed-Muller codes, and obtain nearly-tight bounds. 2. We show that in order to achieve low query complexity, it is sufficient to design one-sided non-adaptive canonical tests. This result is akin to, and based on an analogous result for error-correcting codes due to Ben-Sasson et al. (SIAM J. Computing 35(1) pp1-21)

    MDS array codes for correcting a signle criss-cross error

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    We present a family of maximum-distance separable (MDS) array codes of size (p-1)×(p-1), p a prime number, and minimum criss-cross distance 3, i.e., the code is capable of correcting any row or column in error, without a priori knowledge of what type of error occurred. The complexity of the encoding and decoding algorithms is lower than that of known codes with the same error-correcting power, since our algorithms are based on exclusive-OR operations over lines of different slopes, as opposed to algebraic operations over a finite field. We also provide efficient encoding and decoding algorithms for errors and erasures

    Update-Efficient Regenerating Codes with Minimum Per-Node Storage

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    Regenerating codes provide an efficient way to recover data at failed nodes in distributed storage systems. It has been shown that regenerating codes can be designed to minimize the per-node storage (called MSR) or minimize the communication overhead for regeneration (called MBR). In this work, we propose a new encoding scheme for [n,d] error- correcting MSR codes that generalizes our earlier work on error-correcting regenerating codes. We show that by choosing a suitable diagonal matrix, any generator matrix of the [n,{\alpha}] Reed-Solomon (RS) code can be integrated into the encoding matrix. Hence, MSR codes with the least update complexity can be found. An efficient decoding scheme is also proposed that utilizes the [n,{\alpha}] RS code to perform data reconstruction. The proposed decoding scheme has better error correction capability and incurs the least number of node accesses when errors are present.Comment: Submitted to IEEE ISIT 201

    Concatenated Quantum Codes Constructible in Polynomial Time: Efficient Decoding and Error Correction

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    A method for concatenating quantum error-correcting codes is presented. The method is applicable to a wide class of quantum error-correcting codes known as Calderbank-Shor-Steane (CSS) codes. As a result, codes that achieve a high rate in the Shannon theoretic sense and that are decodable in polynomial time are presented. The rate is the highest among those known to be achievable by CSS codes. Moreover, the best known lower bound on the greatest minimum distance of codes constructible in polynomial time is improved for a wide range.Comment: 16 pages, 3 figures. Ver.4: Title changed. Ver.3: Due to a request of the AE of the journal, the present version has become a combination of (thoroughly revised) quant-ph/0610194 and the former quant-ph/0610195. Problem formulations of polynomial complexity are strictly followed. An erroneous instance of a lower bound on minimum distance was remove

    Improved Decoding of Staircase Codes: The Soft-aided Bit-marking (SABM) Algorithm

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    Staircase codes (SCCs) are typically decoded using iterative bounded-distance decoding (BDD) and hard decisions. In this paper, a novel decoding algorithm is proposed, which partially uses soft information from the channel. The proposed algorithm is based on marking certain number of highly reliable and highly unreliable bits. These marked bits are used to improve the miscorrection-detection capability of the SCC decoder and the error-correcting capability of BDD. For SCCs with 22-error-correcting Bose-Chaudhuri-Hocquenghem component codes, our algorithm improves upon standard SCC decoding by up to 0.300.30~dB at a bit-error rate (BER) of 10710^{-7}. The proposed algorithm is shown to achieve almost half of the gain achievable by an idealized decoder with this structure. A complexity analysis based on the number of additional calls to the component BDD decoder shows that the relative complexity increase is only around 4%4\% at a BER of 10410^{-4}. This additional complexity is shown to decrease as the channel quality improves. Our algorithm is also extended (with minor modifications) to product codes. The simulation results show that in this case, the algorithm offers gains of up to 0.440.44~dB at a BER of 10810^{-8}.Comment: 10 pages, 12 figure
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